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Polytope of Type {4,27,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,27,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,20783)
Rank : 4
Schlafli Type : {4,27,4}
Number of vertices, edges, etc : 8, 108, 108, 4
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,27,4}*864
   3-fold quotients : {4,9,4}*576b
   4-fold quotients : {2,27,4}*432
   6-fold quotients : {4,9,4}*288
   9-fold quotients : {4,3,4}*192b
   12-fold quotients : {2,9,4}*144
   18-fold quotients : {4,3,4}*96
   36-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,441)(  2,442)(  3,443)(  4,444)(  5,445)(  6,446)(  7,447)(  8,448)
(  9,433)( 10,434)( 11,435)( 12,436)( 13,437)( 14,438)( 15,439)( 16,440)
( 17,457)( 18,458)( 19,459)( 20,460)( 21,461)( 22,462)( 23,463)( 24,464)
( 25,449)( 26,450)( 27,451)( 28,452)( 29,453)( 30,454)( 31,455)( 32,456)
( 33,473)( 34,474)( 35,475)( 36,476)( 37,477)( 38,478)( 39,479)( 40,480)
( 41,465)( 42,466)( 43,467)( 44,468)( 45,469)( 46,470)( 47,471)( 48,472)
( 49,489)( 50,490)( 51,491)( 52,492)( 53,493)( 54,494)( 55,495)( 56,496)
( 57,481)( 58,482)( 59,483)( 60,484)( 61,485)( 62,486)( 63,487)( 64,488)
( 65,505)( 66,506)( 67,507)( 68,508)( 69,509)( 70,510)( 71,511)( 72,512)
( 73,497)( 74,498)( 75,499)( 76,500)( 77,501)( 78,502)( 79,503)( 80,504)
( 81,521)( 82,522)( 83,523)( 84,524)( 85,525)( 86,526)( 87,527)( 88,528)
( 89,513)( 90,514)( 91,515)( 92,516)( 93,517)( 94,518)( 95,519)( 96,520)
( 97,537)( 98,538)( 99,539)(100,540)(101,541)(102,542)(103,543)(104,544)
(105,529)(106,530)(107,531)(108,532)(109,533)(110,534)(111,535)(112,536)
(113,553)(114,554)(115,555)(116,556)(117,557)(118,558)(119,559)(120,560)
(121,545)(122,546)(123,547)(124,548)(125,549)(126,550)(127,551)(128,552)
(129,569)(130,570)(131,571)(132,572)(133,573)(134,574)(135,575)(136,576)
(137,561)(138,562)(139,563)(140,564)(141,565)(142,566)(143,567)(144,568)
(145,585)(146,586)(147,587)(148,588)(149,589)(150,590)(151,591)(152,592)
(153,577)(154,578)(155,579)(156,580)(157,581)(158,582)(159,583)(160,584)
(161,601)(162,602)(163,603)(164,604)(165,605)(166,606)(167,607)(168,608)
(169,593)(170,594)(171,595)(172,596)(173,597)(174,598)(175,599)(176,600)
(177,617)(178,618)(179,619)(180,620)(181,621)(182,622)(183,623)(184,624)
(185,609)(186,610)(187,611)(188,612)(189,613)(190,614)(191,615)(192,616)
(193,633)(194,634)(195,635)(196,636)(197,637)(198,638)(199,639)(200,640)
(201,625)(202,626)(203,627)(204,628)(205,629)(206,630)(207,631)(208,632)
(209,649)(210,650)(211,651)(212,652)(213,653)(214,654)(215,655)(216,656)
(217,641)(218,642)(219,643)(220,644)(221,645)(222,646)(223,647)(224,648)
(225,665)(226,666)(227,667)(228,668)(229,669)(230,670)(231,671)(232,672)
(233,657)(234,658)(235,659)(236,660)(237,661)(238,662)(239,663)(240,664)
(241,681)(242,682)(243,683)(244,684)(245,685)(246,686)(247,687)(248,688)
(249,673)(250,674)(251,675)(252,676)(253,677)(254,678)(255,679)(256,680)
(257,697)(258,698)(259,699)(260,700)(261,701)(262,702)(263,703)(264,704)
(265,689)(266,690)(267,691)(268,692)(269,693)(270,694)(271,695)(272,696)
(273,713)(274,714)(275,715)(276,716)(277,717)(278,718)(279,719)(280,720)
(281,705)(282,706)(283,707)(284,708)(285,709)(286,710)(287,711)(288,712)
(289,729)(290,730)(291,731)(292,732)(293,733)(294,734)(295,735)(296,736)
(297,721)(298,722)(299,723)(300,724)(301,725)(302,726)(303,727)(304,728)
(305,745)(306,746)(307,747)(308,748)(309,749)(310,750)(311,751)(312,752)
(313,737)(314,738)(315,739)(316,740)(317,741)(318,742)(319,743)(320,744)
(321,761)(322,762)(323,763)(324,764)(325,765)(326,766)(327,767)(328,768)
(329,753)(330,754)(331,755)(332,756)(333,757)(334,758)(335,759)(336,760)
(337,777)(338,778)(339,779)(340,780)(341,781)(342,782)(343,783)(344,784)
(345,769)(346,770)(347,771)(348,772)(349,773)(350,774)(351,775)(352,776)
(353,793)(354,794)(355,795)(356,796)(357,797)(358,798)(359,799)(360,800)
(361,785)(362,786)(363,787)(364,788)(365,789)(366,790)(367,791)(368,792)
(369,809)(370,810)(371,811)(372,812)(373,813)(374,814)(375,815)(376,816)
(377,801)(378,802)(379,803)(380,804)(381,805)(382,806)(383,807)(384,808)
(385,825)(386,826)(387,827)(388,828)(389,829)(390,830)(391,831)(392,832)
(393,817)(394,818)(395,819)(396,820)(397,821)(398,822)(399,823)(400,824)
(401,841)(402,842)(403,843)(404,844)(405,845)(406,846)(407,847)(408,848)
(409,833)(410,834)(411,835)(412,836)(413,837)(414,838)(415,839)(416,840)
(417,857)(418,858)(419,859)(420,860)(421,861)(422,862)(423,863)(424,864)
(425,849)(426,850)(427,851)(428,852)(429,853)(430,854)(431,855)(432,856);;
s1 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)( 50,130)
( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)( 58,142)
( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)( 66,114)
( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)( 74,126)
( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)( 82, 98)
( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)(146,418)
(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)(154,430)
(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)(162,402)
(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)(170,414)
(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)(178,386)
(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)(186,398)
(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)(194,370)
(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)(202,382)
(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)(210,354)
(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)(218,366)
(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)(226,338)
(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)(234,350)
(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)(242,322)
(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)(250,334)
(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)(258,306)
(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)(266,318)
(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)(274,290)
(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)(282,302)
(283,304)(284,303)(285,297)(286,298)(287,300)(288,299)(435,436)(439,440)
(441,445)(442,446)(443,448)(444,447)(449,465)(450,466)(451,468)(452,467)
(453,469)(454,470)(455,472)(456,471)(457,477)(458,478)(459,480)(460,479)
(461,473)(462,474)(463,476)(464,475)(481,561)(482,562)(483,564)(484,563)
(485,565)(486,566)(487,568)(488,567)(489,573)(490,574)(491,576)(492,575)
(493,569)(494,570)(495,572)(496,571)(497,545)(498,546)(499,548)(500,547)
(501,549)(502,550)(503,552)(504,551)(505,557)(506,558)(507,560)(508,559)
(509,553)(510,554)(511,556)(512,555)(513,529)(514,530)(515,532)(516,531)
(517,533)(518,534)(519,536)(520,535)(521,541)(522,542)(523,544)(524,543)
(525,537)(526,538)(527,540)(528,539)(577,849)(578,850)(579,852)(580,851)
(581,853)(582,854)(583,856)(584,855)(585,861)(586,862)(587,864)(588,863)
(589,857)(590,858)(591,860)(592,859)(593,833)(594,834)(595,836)(596,835)
(597,837)(598,838)(599,840)(600,839)(601,845)(602,846)(603,848)(604,847)
(605,841)(606,842)(607,844)(608,843)(609,817)(610,818)(611,820)(612,819)
(613,821)(614,822)(615,824)(616,823)(617,829)(618,830)(619,832)(620,831)
(621,825)(622,826)(623,828)(624,827)(625,801)(626,802)(627,804)(628,803)
(629,805)(630,806)(631,808)(632,807)(633,813)(634,814)(635,816)(636,815)
(637,809)(638,810)(639,812)(640,811)(641,785)(642,786)(643,788)(644,787)
(645,789)(646,790)(647,792)(648,791)(649,797)(650,798)(651,800)(652,799)
(653,793)(654,794)(655,796)(656,795)(657,769)(658,770)(659,772)(660,771)
(661,773)(662,774)(663,776)(664,775)(665,781)(666,782)(667,784)(668,783)
(669,777)(670,778)(671,780)(672,779)(673,753)(674,754)(675,756)(676,755)
(677,757)(678,758)(679,760)(680,759)(681,765)(682,766)(683,768)(684,767)
(685,761)(686,762)(687,764)(688,763)(689,737)(690,738)(691,740)(692,739)
(693,741)(694,742)(695,744)(696,743)(697,749)(698,750)(699,752)(700,751)
(701,745)(702,746)(703,748)(704,747)(705,721)(706,722)(707,724)(708,723)
(709,725)(710,726)(711,728)(712,727)(713,733)(714,734)(715,736)(716,735)
(717,729)(718,730)(719,732)(720,731);;
s2 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,157)(  6,160)(  7,159)(  8,158)
(  9,153)( 10,156)( 11,155)( 12,154)( 13,149)( 14,152)( 15,151)( 16,150)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,189)( 22,192)( 23,191)( 24,190)
( 25,185)( 26,188)( 27,187)( 28,186)( 29,181)( 30,184)( 31,183)( 32,182)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,173)( 38,176)( 39,175)( 40,174)
( 41,169)( 42,172)( 43,171)( 44,170)( 45,165)( 46,168)( 47,167)( 48,166)
( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)( 56,286)
( 57,281)( 58,284)( 59,283)( 60,282)( 61,277)( 62,280)( 63,279)( 64,278)
( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)( 72,270)
( 73,265)( 74,268)( 75,267)( 76,266)( 77,261)( 78,264)( 79,263)( 80,262)
( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)( 88,254)
( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)( 96,246)
( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)(104,238)
(105,233)(106,236)(107,235)(108,234)(109,229)(110,232)(111,231)(112,230)
(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)(120,222)
(121,217)(122,220)(123,219)(124,218)(125,213)(126,216)(127,215)(128,214)
(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)(136,206)
(137,201)(138,204)(139,203)(140,202)(141,197)(142,200)(143,199)(144,198)
(289,417)(290,420)(291,419)(292,418)(293,429)(294,432)(295,431)(296,430)
(297,425)(298,428)(299,427)(300,426)(301,421)(302,424)(303,423)(304,422)
(305,401)(306,404)(307,403)(308,402)(309,413)(310,416)(311,415)(312,414)
(313,409)(314,412)(315,411)(316,410)(317,405)(318,408)(319,407)(320,406)
(321,385)(322,388)(323,387)(324,386)(325,397)(326,400)(327,399)(328,398)
(329,393)(330,396)(331,395)(332,394)(333,389)(334,392)(335,391)(336,390)
(337,369)(338,372)(339,371)(340,370)(341,381)(342,384)(343,383)(344,382)
(345,377)(346,380)(347,379)(348,378)(349,373)(350,376)(351,375)(352,374)
(354,356)(357,365)(358,368)(359,367)(360,366)(362,364)(433,577)(434,580)
(435,579)(436,578)(437,589)(438,592)(439,591)(440,590)(441,585)(442,588)
(443,587)(444,586)(445,581)(446,584)(447,583)(448,582)(449,609)(450,612)
(451,611)(452,610)(453,621)(454,624)(455,623)(456,622)(457,617)(458,620)
(459,619)(460,618)(461,613)(462,616)(463,615)(464,614)(465,593)(466,596)
(467,595)(468,594)(469,605)(470,608)(471,607)(472,606)(473,601)(474,604)
(475,603)(476,602)(477,597)(478,600)(479,599)(480,598)(481,705)(482,708)
(483,707)(484,706)(485,717)(486,720)(487,719)(488,718)(489,713)(490,716)
(491,715)(492,714)(493,709)(494,712)(495,711)(496,710)(497,689)(498,692)
(499,691)(500,690)(501,701)(502,704)(503,703)(504,702)(505,697)(506,700)
(507,699)(508,698)(509,693)(510,696)(511,695)(512,694)(513,673)(514,676)
(515,675)(516,674)(517,685)(518,688)(519,687)(520,686)(521,681)(522,684)
(523,683)(524,682)(525,677)(526,680)(527,679)(528,678)(529,657)(530,660)
(531,659)(532,658)(533,669)(534,672)(535,671)(536,670)(537,665)(538,668)
(539,667)(540,666)(541,661)(542,664)(543,663)(544,662)(545,641)(546,644)
(547,643)(548,642)(549,653)(550,656)(551,655)(552,654)(553,649)(554,652)
(555,651)(556,650)(557,645)(558,648)(559,647)(560,646)(561,625)(562,628)
(563,627)(564,626)(565,637)(566,640)(567,639)(568,638)(569,633)(570,636)
(571,635)(572,634)(573,629)(574,632)(575,631)(576,630)(721,849)(722,852)
(723,851)(724,850)(725,861)(726,864)(727,863)(728,862)(729,857)(730,860)
(731,859)(732,858)(733,853)(734,856)(735,855)(736,854)(737,833)(738,836)
(739,835)(740,834)(741,845)(742,848)(743,847)(744,846)(745,841)(746,844)
(747,843)(748,842)(749,837)(750,840)(751,839)(752,838)(753,817)(754,820)
(755,819)(756,818)(757,829)(758,832)(759,831)(760,830)(761,825)(762,828)
(763,827)(764,826)(765,821)(766,824)(767,823)(768,822)(769,801)(770,804)
(771,803)(772,802)(773,813)(774,816)(775,815)(776,814)(777,809)(778,812)
(779,811)(780,810)(781,805)(782,808)(783,807)(784,806)(786,788)(789,797)
(790,800)(791,799)(792,798)(794,796);;
s3 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)
(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)
(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)
(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)
(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)
(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)
(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432)
(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)(447,448)
(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)(463,464)
(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)(479,480)
(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)(495,496)
(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)(511,512)
(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)(527,528)
(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)(543,544)
(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)(559,560)
(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)(575,576)
(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)(591,592)
(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)(607,608)
(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)(623,624)
(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)(639,640)
(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)(655,656)
(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)(671,672)
(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)(687,688)
(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)(703,704)
(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)(719,720)
(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)(735,736)
(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)(751,752)
(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)(767,768)
(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)(783,784)
(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)(799,800)
(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)(815,816)
(817,818)(819,820)(821,822)(823,824)(825,826)(827,828)(829,830)(831,832)
(833,834)(835,836)(837,838)(839,840)(841,842)(843,844)(845,846)(847,848)
(849,850)(851,852)(853,854)(855,856)(857,858)(859,860)(861,862)(863,864);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,441)(  2,442)(  3,443)(  4,444)(  5,445)(  6,446)(  7,447)
(  8,448)(  9,433)( 10,434)( 11,435)( 12,436)( 13,437)( 14,438)( 15,439)
( 16,440)( 17,457)( 18,458)( 19,459)( 20,460)( 21,461)( 22,462)( 23,463)
( 24,464)( 25,449)( 26,450)( 27,451)( 28,452)( 29,453)( 30,454)( 31,455)
( 32,456)( 33,473)( 34,474)( 35,475)( 36,476)( 37,477)( 38,478)( 39,479)
( 40,480)( 41,465)( 42,466)( 43,467)( 44,468)( 45,469)( 46,470)( 47,471)
( 48,472)( 49,489)( 50,490)( 51,491)( 52,492)( 53,493)( 54,494)( 55,495)
( 56,496)( 57,481)( 58,482)( 59,483)( 60,484)( 61,485)( 62,486)( 63,487)
( 64,488)( 65,505)( 66,506)( 67,507)( 68,508)( 69,509)( 70,510)( 71,511)
( 72,512)( 73,497)( 74,498)( 75,499)( 76,500)( 77,501)( 78,502)( 79,503)
( 80,504)( 81,521)( 82,522)( 83,523)( 84,524)( 85,525)( 86,526)( 87,527)
( 88,528)( 89,513)( 90,514)( 91,515)( 92,516)( 93,517)( 94,518)( 95,519)
( 96,520)( 97,537)( 98,538)( 99,539)(100,540)(101,541)(102,542)(103,543)
(104,544)(105,529)(106,530)(107,531)(108,532)(109,533)(110,534)(111,535)
(112,536)(113,553)(114,554)(115,555)(116,556)(117,557)(118,558)(119,559)
(120,560)(121,545)(122,546)(123,547)(124,548)(125,549)(126,550)(127,551)
(128,552)(129,569)(130,570)(131,571)(132,572)(133,573)(134,574)(135,575)
(136,576)(137,561)(138,562)(139,563)(140,564)(141,565)(142,566)(143,567)
(144,568)(145,585)(146,586)(147,587)(148,588)(149,589)(150,590)(151,591)
(152,592)(153,577)(154,578)(155,579)(156,580)(157,581)(158,582)(159,583)
(160,584)(161,601)(162,602)(163,603)(164,604)(165,605)(166,606)(167,607)
(168,608)(169,593)(170,594)(171,595)(172,596)(173,597)(174,598)(175,599)
(176,600)(177,617)(178,618)(179,619)(180,620)(181,621)(182,622)(183,623)
(184,624)(185,609)(186,610)(187,611)(188,612)(189,613)(190,614)(191,615)
(192,616)(193,633)(194,634)(195,635)(196,636)(197,637)(198,638)(199,639)
(200,640)(201,625)(202,626)(203,627)(204,628)(205,629)(206,630)(207,631)
(208,632)(209,649)(210,650)(211,651)(212,652)(213,653)(214,654)(215,655)
(216,656)(217,641)(218,642)(219,643)(220,644)(221,645)(222,646)(223,647)
(224,648)(225,665)(226,666)(227,667)(228,668)(229,669)(230,670)(231,671)
(232,672)(233,657)(234,658)(235,659)(236,660)(237,661)(238,662)(239,663)
(240,664)(241,681)(242,682)(243,683)(244,684)(245,685)(246,686)(247,687)
(248,688)(249,673)(250,674)(251,675)(252,676)(253,677)(254,678)(255,679)
(256,680)(257,697)(258,698)(259,699)(260,700)(261,701)(262,702)(263,703)
(264,704)(265,689)(266,690)(267,691)(268,692)(269,693)(270,694)(271,695)
(272,696)(273,713)(274,714)(275,715)(276,716)(277,717)(278,718)(279,719)
(280,720)(281,705)(282,706)(283,707)(284,708)(285,709)(286,710)(287,711)
(288,712)(289,729)(290,730)(291,731)(292,732)(293,733)(294,734)(295,735)
(296,736)(297,721)(298,722)(299,723)(300,724)(301,725)(302,726)(303,727)
(304,728)(305,745)(306,746)(307,747)(308,748)(309,749)(310,750)(311,751)
(312,752)(313,737)(314,738)(315,739)(316,740)(317,741)(318,742)(319,743)
(320,744)(321,761)(322,762)(323,763)(324,764)(325,765)(326,766)(327,767)
(328,768)(329,753)(330,754)(331,755)(332,756)(333,757)(334,758)(335,759)
(336,760)(337,777)(338,778)(339,779)(340,780)(341,781)(342,782)(343,783)
(344,784)(345,769)(346,770)(347,771)(348,772)(349,773)(350,774)(351,775)
(352,776)(353,793)(354,794)(355,795)(356,796)(357,797)(358,798)(359,799)
(360,800)(361,785)(362,786)(363,787)(364,788)(365,789)(366,790)(367,791)
(368,792)(369,809)(370,810)(371,811)(372,812)(373,813)(374,814)(375,815)
(376,816)(377,801)(378,802)(379,803)(380,804)(381,805)(382,806)(383,807)
(384,808)(385,825)(386,826)(387,827)(388,828)(389,829)(390,830)(391,831)
(392,832)(393,817)(394,818)(395,819)(396,820)(397,821)(398,822)(399,823)
(400,824)(401,841)(402,842)(403,843)(404,844)(405,845)(406,846)(407,847)
(408,848)(409,833)(410,834)(411,835)(412,836)(413,837)(414,838)(415,839)
(416,840)(417,857)(418,858)(419,859)(420,860)(421,861)(422,862)(423,863)
(424,864)(425,849)(426,850)(427,851)(428,852)(429,853)(430,854)(431,855)
(432,856);
s1 := Sym(864)!(  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)
( 50,130)( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)
( 58,142)( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)
( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)
( 74,126)( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)
( 82, 98)( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)
(146,418)(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)
(154,430)(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)
(162,402)(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)
(170,414)(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)
(178,386)(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)
(186,398)(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)
(194,370)(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)
(202,382)(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)
(210,354)(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)
(218,366)(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)
(226,338)(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)
(234,350)(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)
(242,322)(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)
(250,334)(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)
(258,306)(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)
(266,318)(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)
(274,290)(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)
(282,302)(283,304)(284,303)(285,297)(286,298)(287,300)(288,299)(435,436)
(439,440)(441,445)(442,446)(443,448)(444,447)(449,465)(450,466)(451,468)
(452,467)(453,469)(454,470)(455,472)(456,471)(457,477)(458,478)(459,480)
(460,479)(461,473)(462,474)(463,476)(464,475)(481,561)(482,562)(483,564)
(484,563)(485,565)(486,566)(487,568)(488,567)(489,573)(490,574)(491,576)
(492,575)(493,569)(494,570)(495,572)(496,571)(497,545)(498,546)(499,548)
(500,547)(501,549)(502,550)(503,552)(504,551)(505,557)(506,558)(507,560)
(508,559)(509,553)(510,554)(511,556)(512,555)(513,529)(514,530)(515,532)
(516,531)(517,533)(518,534)(519,536)(520,535)(521,541)(522,542)(523,544)
(524,543)(525,537)(526,538)(527,540)(528,539)(577,849)(578,850)(579,852)
(580,851)(581,853)(582,854)(583,856)(584,855)(585,861)(586,862)(587,864)
(588,863)(589,857)(590,858)(591,860)(592,859)(593,833)(594,834)(595,836)
(596,835)(597,837)(598,838)(599,840)(600,839)(601,845)(602,846)(603,848)
(604,847)(605,841)(606,842)(607,844)(608,843)(609,817)(610,818)(611,820)
(612,819)(613,821)(614,822)(615,824)(616,823)(617,829)(618,830)(619,832)
(620,831)(621,825)(622,826)(623,828)(624,827)(625,801)(626,802)(627,804)
(628,803)(629,805)(630,806)(631,808)(632,807)(633,813)(634,814)(635,816)
(636,815)(637,809)(638,810)(639,812)(640,811)(641,785)(642,786)(643,788)
(644,787)(645,789)(646,790)(647,792)(648,791)(649,797)(650,798)(651,800)
(652,799)(653,793)(654,794)(655,796)(656,795)(657,769)(658,770)(659,772)
(660,771)(661,773)(662,774)(663,776)(664,775)(665,781)(666,782)(667,784)
(668,783)(669,777)(670,778)(671,780)(672,779)(673,753)(674,754)(675,756)
(676,755)(677,757)(678,758)(679,760)(680,759)(681,765)(682,766)(683,768)
(684,767)(685,761)(686,762)(687,764)(688,763)(689,737)(690,738)(691,740)
(692,739)(693,741)(694,742)(695,744)(696,743)(697,749)(698,750)(699,752)
(700,751)(701,745)(702,746)(703,748)(704,747)(705,721)(706,722)(707,724)
(708,723)(709,725)(710,726)(711,728)(712,727)(713,733)(714,734)(715,736)
(716,735)(717,729)(718,730)(719,732)(720,731);
s2 := Sym(864)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,157)(  6,160)(  7,159)
(  8,158)(  9,153)( 10,156)( 11,155)( 12,154)( 13,149)( 14,152)( 15,151)
( 16,150)( 17,177)( 18,180)( 19,179)( 20,178)( 21,189)( 22,192)( 23,191)
( 24,190)( 25,185)( 26,188)( 27,187)( 28,186)( 29,181)( 30,184)( 31,183)
( 32,182)( 33,161)( 34,164)( 35,163)( 36,162)( 37,173)( 38,176)( 39,175)
( 40,174)( 41,169)( 42,172)( 43,171)( 44,170)( 45,165)( 46,168)( 47,167)
( 48,166)( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)
( 56,286)( 57,281)( 58,284)( 59,283)( 60,282)( 61,277)( 62,280)( 63,279)
( 64,278)( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)
( 72,270)( 73,265)( 74,268)( 75,267)( 76,266)( 77,261)( 78,264)( 79,263)
( 80,262)( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)
( 88,254)( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)
( 96,246)( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)
(104,238)(105,233)(106,236)(107,235)(108,234)(109,229)(110,232)(111,231)
(112,230)(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)
(120,222)(121,217)(122,220)(123,219)(124,218)(125,213)(126,216)(127,215)
(128,214)(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)
(136,206)(137,201)(138,204)(139,203)(140,202)(141,197)(142,200)(143,199)
(144,198)(289,417)(290,420)(291,419)(292,418)(293,429)(294,432)(295,431)
(296,430)(297,425)(298,428)(299,427)(300,426)(301,421)(302,424)(303,423)
(304,422)(305,401)(306,404)(307,403)(308,402)(309,413)(310,416)(311,415)
(312,414)(313,409)(314,412)(315,411)(316,410)(317,405)(318,408)(319,407)
(320,406)(321,385)(322,388)(323,387)(324,386)(325,397)(326,400)(327,399)
(328,398)(329,393)(330,396)(331,395)(332,394)(333,389)(334,392)(335,391)
(336,390)(337,369)(338,372)(339,371)(340,370)(341,381)(342,384)(343,383)
(344,382)(345,377)(346,380)(347,379)(348,378)(349,373)(350,376)(351,375)
(352,374)(354,356)(357,365)(358,368)(359,367)(360,366)(362,364)(433,577)
(434,580)(435,579)(436,578)(437,589)(438,592)(439,591)(440,590)(441,585)
(442,588)(443,587)(444,586)(445,581)(446,584)(447,583)(448,582)(449,609)
(450,612)(451,611)(452,610)(453,621)(454,624)(455,623)(456,622)(457,617)
(458,620)(459,619)(460,618)(461,613)(462,616)(463,615)(464,614)(465,593)
(466,596)(467,595)(468,594)(469,605)(470,608)(471,607)(472,606)(473,601)
(474,604)(475,603)(476,602)(477,597)(478,600)(479,599)(480,598)(481,705)
(482,708)(483,707)(484,706)(485,717)(486,720)(487,719)(488,718)(489,713)
(490,716)(491,715)(492,714)(493,709)(494,712)(495,711)(496,710)(497,689)
(498,692)(499,691)(500,690)(501,701)(502,704)(503,703)(504,702)(505,697)
(506,700)(507,699)(508,698)(509,693)(510,696)(511,695)(512,694)(513,673)
(514,676)(515,675)(516,674)(517,685)(518,688)(519,687)(520,686)(521,681)
(522,684)(523,683)(524,682)(525,677)(526,680)(527,679)(528,678)(529,657)
(530,660)(531,659)(532,658)(533,669)(534,672)(535,671)(536,670)(537,665)
(538,668)(539,667)(540,666)(541,661)(542,664)(543,663)(544,662)(545,641)
(546,644)(547,643)(548,642)(549,653)(550,656)(551,655)(552,654)(553,649)
(554,652)(555,651)(556,650)(557,645)(558,648)(559,647)(560,646)(561,625)
(562,628)(563,627)(564,626)(565,637)(566,640)(567,639)(568,638)(569,633)
(570,636)(571,635)(572,634)(573,629)(574,632)(575,631)(576,630)(721,849)
(722,852)(723,851)(724,850)(725,861)(726,864)(727,863)(728,862)(729,857)
(730,860)(731,859)(732,858)(733,853)(734,856)(735,855)(736,854)(737,833)
(738,836)(739,835)(740,834)(741,845)(742,848)(743,847)(744,846)(745,841)
(746,844)(747,843)(748,842)(749,837)(750,840)(751,839)(752,838)(753,817)
(754,820)(755,819)(756,818)(757,829)(758,832)(759,831)(760,830)(761,825)
(762,828)(763,827)(764,826)(765,821)(766,824)(767,823)(768,822)(769,801)
(770,804)(771,803)(772,802)(773,813)(774,816)(775,815)(776,814)(777,809)
(778,812)(779,811)(780,810)(781,805)(782,808)(783,807)(784,806)(786,788)
(789,797)(790,800)(791,799)(792,798)(794,796);
s3 := Sym(864)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)
(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)
(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)
(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)
(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)
(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)
(431,432)(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)
(447,448)(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)
(463,464)(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)
(479,480)(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)
(495,496)(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)
(511,512)(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)
(527,528)(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)
(543,544)(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)
(559,560)(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)
(575,576)(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)
(591,592)(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)
(607,608)(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)
(623,624)(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)
(639,640)(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)
(655,656)(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)
(671,672)(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)
(687,688)(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)
(703,704)(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)
(719,720)(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)
(735,736)(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)
(751,752)(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)
(767,768)(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)
(783,784)(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)
(799,800)(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)
(815,816)(817,818)(819,820)(821,822)(823,824)(825,826)(827,828)(829,830)
(831,832)(833,834)(835,836)(837,838)(839,840)(841,842)(843,844)(845,846)
(847,848)(849,850)(851,852)(853,854)(855,856)(857,858)(859,860)(861,862)
(863,864);
poly := sub<Sym(864)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope